Local gauge invariance

Starting from a completely different angle, namely, the nuclear Lagrangean and the requirement of local gauge invariance, we have shown in Section IV.B... 

The derivation of Eq. (218) from Eq. (206) follows from local gauge invariance, and it is always possible to apply a local gauge transform to the vector A, the Maxwell vector potential. The ordinary derivative of the d Alembert wave equation is replaced by an 0(3) covariant derivative. The U(l) equivalent of Eq. (218) in quantum-mechanical (operator) form is Eq. (13), and Eq. (212) is the rigorously correct form of the phenomenological Eq. (25). It can be seen that Eq. (212) is richly structured in the vacuum and must be solved numerically. The vacuum currents present in Eq. (218) can be computed from the right-hand side of the wave equation (212), and these vacuum currents follow from local gauge invariance. 

The locally gauge-invariant Lehnert field equation corresponding to Eq. (374) was derived as Eq. (350). The photon picks up mass from the vacuum itself, and having derived a locally gauge-invariant Proca equation, canonical quantization can be applied to produce a photon with mass with three space dimensions. 

To an excellent approximation, the four Klein-Gordon equations (443) are d Alembert equations, which are locally gauge-invariant. 

However, there remains the problem of how to obtain a locally gauge-invariant Proca equation. To address this problem rigorously, it is necessary to use a non-Abelian Higgs mechanism applied within gauge theory. 

At the Higgs minimum, this field equation reduces to the locally gauge-invariant Proca equation... 

Beyond atomic spectroscopy muonium renders the possibility to search directly and sensitively for yet unknown interactions between the two charged leptons from two different generations. Among the mysteries observed for leptons are the apparently conserved lepton numbers. As a matter of fact, several distinctively different lepton number conservation schemes appear to hold, some of which are additive and some are multiplicative, parity-like. Some of them distinguish between lepton families and others don t [46,47,48,49,50]. No local gauge invariance has been revealed yet which would be associated with any of these empirically established laws. Since there is common believe [51] that any discrete conserved quantity is connected to a local gauge invariance, a breakdown of lepton number conservation is widely expected, particularly in the framework of many speculative models. 

An immediate consequence of the local gauge invariance of the Lagrangian is current conservation,... 

The principle of local invariance in a curved Riemannian manifold leads to the appearance of compensating fields. Like the electromagnetic field, which is the compensating field of local phase transformation, the gravitational field may be interpreted as the compensating field of local Lorentz transformations. In modern physics all interactions are understood in terms of theories which combine local gauge invariance with spontaneous symmetry breaking. 

Electrodynamics is locally gauge invariant because all derivatives occur in special combinations D, called covariant derivatives which do have the property that... 

Non-Abelian local gauge invariance—Yang-Mills theories... 

We now turn to the much more subtle question of local non-Abelian gauge invariance. The first generalization of SU 2) to locally gauge invariant Lagrangians is due to Yang and Mills (1954) [a detailed account can be found in Taylor (1976)], but the treatment applies to any group with a finite number of generators [see Abers and Lee (1973)]. 

According to Section 2.3 the locally gauge invariant Lagrangian for the coupling of the gauge bosons (GB) to the scalars (5) is... 

Apart from the above symmetry considerations the SB functional-integral formalism reveals additional global and local gauge invariances [25, 30-32]. 

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